Eliashberg Theory: a short review

F. MarsiglioDepartment of Physics, University of Alberta, Edmonton, AB, Canada T6G 2E1

(Dated: November 13, 2019)

Eliashberg theory is a theory of superconductivity that describes the role of phonons in providingthe attractive interaction between two electrons. Phonon dynamics are taken into account, thusgiving rise to retardation effects that impact the electrons, in the form of a frequency-dependentelectron self-energy. In the superconducting state, this means that the order parameter, generallyconsidered to be a static quantity in the Bardeen-Cooper-Schrieffer (BCS) theory, also becomesfrequency dependent. Here we review the finite temperature formulation of Eliashberg theory, bothon the imaginary and real frequency axis, and briefly display some examples of the consequences ofa dynamical, as opposed to static, interaction. Along the way we point out where further work isrequired, concerning the validity of some of the assumptions used.

I. INTRODUCTION

Superconductivity is a remarkable phenomenon, notleast because it represents a manifestation of the quan-tum world on a macroscopic scale. It is spectacularlydemonstrated with levitating train sets,1 and indeed thisproperty and many others of superconductors are slowlybeing utilized in everyday applications.2 However, the es-tablished practice of incorporating superconductors intothe real world should not be taken as an indication that“the last nail in the coffin [of superconductivity]”3 hasbeen achieved. On the contrary, in the intervening half-century since this quote was written, many new super-conductors have been discovered, and we have reached apoint where it is clear that a deep lack of understanding4

of superconductivity currently exists. Reference [5] com-piles a series of articles reviewing the various “families”or classes of superconductors, where one can readily seecommon and different characteristics. At the momentmany of these classes require a class-specific mechanismfor superconductivity, a clearly untenable situation, inmy opinion. Further classes have been discovered or ex-panded upon since, such as nickelates,7 and the hydridesunder pressure,6 for example.

Our “deep lack of understanding” should not be takento indicate that theoretical contributions have not beenforthcoming. In fact there have been remarkable con-tributions to key theoretical ideas in physics that stemfrom research in superconductivity, starting with Londontheory8 and Ginzburg-Landau9 theory, through to BCStheory.10 When Gor’kov11 recast the BCS theory of su-perconductivity in the language of Green functions, thenthe stage was set for Eliashberg12,13 to formulate the the-ory that bears his name. It is fitting that we honour thelasting impact of his work with this brief review, on theoccasion of his 90th birthday, and the 60th anniversaryof the publication of two papers that paved the path forconsiderable future quantitative work in superconductiv-ity. Based on an index I am fond of using for famouspeople, his name appears in titles of papers 248 times,and in abstracts and keywords 1439 times.16

Before proceeding further, we wish to make some re-marks about the nature of this review. It will necessarily

repeat material from previous reviews, which we cata-logue as follows. Scalapino17 and McMillan and Rowell18

perhaps gave one of the first comprehensive reviews ofboth calculations and experiments that provide remark-able evidence for the validity of Eliashberg theory forvarious superconductors. These reviews were providedin the comprehensive monogram by Parks;19 the readershould refer to this monogram and the references therein,as we cannot possibly properly reference all the primaryliterature sources before “Parks”, as this would consumetoo many pages here. The author list in Parks is a who’swho of experts in superconductivity, with two notable ex-ceptions, John Bardeen and Gerasim (Sima) Eliashberg.

A subsequent very influential review was that of Allenand Mitrović,20 where mostly superconducting Tc wasdiscussed. These authors highlighted the expediency ofdoing many calculations on the imaginary frequency axis,a possibility first noted in Ref. [21] and utilized to greatadvantage in subsequent years.22–25

A few years later Rainer wrote a “state-of-the-union”address26 on first principles calculations of superconduct-ing Tc in which a challenge was issued to both bandstructure and many-body theorists. For the former, themissing ingredient was a complete (italics are mine) cal-culation of the electron-phonon coupling. These werefirst calculated in the 1960’s (e.g. Ref. [27]) but haveexperienced vast improvement over the past 50 years,through the adoption and improvement of Density Func-tional Theory methods, plus the increased computationalability achieved in the intervening decades. Excellentsummaries of this progress is provided in Refs. [28,29,30],where two alternative procedures are described. The firstfollows the original route of determining the electron-phonon interaction and including this as input to theEliashberg equations, while the second aims to treatboth the electron-phonon-induced electron-electron in-teraction and the direct Coulomb interaction on an equalfooting. The result is advertised to meet Rainer’s chal-lenge and calculate Tc and other superconducting prop-erties without any experimental input. It is noteworthythat in the second formulation (see also Refs. [31] and[32]) the equations are BCS-like and do not depend onfrequency, but only momentum. In the usual formula-

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tion, the calculation of µ∗, the effective direct electron-electron Coulomb interaction, is often simply assigneda (small) numerical value, and therefore is treated phe-nomenologically as a fitting parameter. The challengeto many-body theorists remains, as more superconduc-tors have been discovered that seem to extend beyondthe weak coupling regime, and likely require descriptionsbeyond BCS and Eliashberg theory.

In 1990 the review by Carbotte33 provided a compre-hensive update for a number of thermodynamic proper-ties of various superconductors known at the time, in-cluding the high temperature cuprate materials. Partlyfor this reason his review is titled “Properties of boson-exchange superconductors,” since there was a feeling atthe time (and still is in parts of the community) that theEliashberg framework might apply to these superconduc-tors, but with exchange of a boson other than the phonon.Quite a few years later we wrote a review jointly,34 fo-cusing on “electron-phonon superconductivity.” This re-view summarized known properties and extended resultsto dynamical properties such as the optical conductivity,building on earlier work in Ref. [35] and mini-reviews inRef. [36]. More recently Ummarino has published a mini-review with some generalizations to multi-band and theiron pnictide superconductors.37

The other remark we should make is that while Eliash-berg theory has been extremely successful, we will alsopoint out the limitations that exist. Indeed, thesewere recognized right from the beginning, with bothEliashberg12 and Migdal38 emphasizing that limitationsexist on the value of the dimensionless coupling parame-ter, λ, due to the expected phonon softening that wouldoccur as λ increases. They claimed an upper limit ofλ ≈ 1, which then significantly restricts the domain ofvalidity of the theory. Constraints on the parameterswould be a constant theme over the ensuing years. In1968 McMillan39 gave more quantitative arguments fora maximum Tc, based on the expected relationship be-tween the coupling strength and the phonon frequency.This was reinforced by Cohen and Anderson40 and hasbeen discussed critically a number of times since.41,42

Alexandrov43 has also raised objections, based on po-laron collapse, a topic we will revisit later.

Some of the early history regarding the origins of theelectron-phonon interaction was provided in Ref. [34]and will be omitted here. By the early to mid 1950’sFröhlich44 and Bardeen and Pines45 had established thatthe effective Hamiltonian for the electron-phonon inter-action had a potential interaction of the form46

V effk,k′ =4πe2

(k− k′)2 + k2TF

[1+

h̄2ω2(k− k′)(�k − �k′)2 − h̄2ω2(k− k′)

],

(1)where kTF is the Thomas–Fermi wave vector, and ω(q)

is the dressed phonon frequency. This part of theHamiltonian represents the pairing interaction betweentwo electrons with wave vectors k and k′ in the FirstBrillouin Zone (FBZ) and energies �k and �k′ . Theinteraction Hamiltonian written in this form is oftensaid to have “the phonons integrated out.” It was onthe basis of this Hamiltonian that Bardeen, Cooper andSchrieffer (BCS)10 formulated a model Hamiltonian withan attractive (negative in sign) interaction for electronenergies near the Fermi energy, �F .

II. THE ELIASHBERG EQUATIONS

Eliashberg,12 following what Migdal38 had calculatedin the normal state, did not “integrate out the phonons,”but instead adopted the Hamiltonian

H =∑kσ

(�k − µ)c†kσckσ +∑q

h̄ωqa†qaq

+1√N

∑kk′σ

g(k,k′)(ak−k′ + a

†−(k−k′)

)c†k′σckσ .(2)

where ckσ (c†kσ) is the annihilation (creation) operator for

an electron with spin σ and wave vector k, and aq (a†q)

is the annihilation (creation) operator for a phonon withwave vector q. The electron-phonon coupling function,g(k,k′) is generally a function of both wave vectors (andnot just their difference), and in principle is calculablewith the Density Functional Theory Methods mentionedearlier. Very often models are adopted based on simple(e.g. tight-binding) considerations.

Eliashberg then applied the apparatus of field theoryto formulate a pairing theory that accounts for the dy-namics of the interaction, i.e. for retardation effects.A sketch of the derivation, taken from Rickayzen47 (seealso Ref. [34]), is provided in the Appendix. This is myfavourite derivation, as it does not rely on a formalism(e.g. the Nambu formalism) whose validity requires anact of faith (or, you simply work through everything any-ways, to ensure that the formalism “works”).

Following Eliashberg13 with more modernnotation,17,20 the “normal” self energy Σ(k, iωm) isseparated out into even and odd (in Matsubara fre-quency) parts, so that two new functions, Z and χ aredefined:

iωm[1− Z(k, iωm)

]≡ 1

2

[Σ(k, iωm)− Σ(k,−iωm)

]χ(k, iωm) ≡

1

2

[Σ(k, iωm) + Σ(k,−iωm)

].(3)

The equations that emerge are

3

Z(k, iωm) = 1 +1

Nβ

∑k′,m′

λkk′(iωm − iωm′)g(�F )

(ωm′/ωm

)Z(k′, iωm′)

ω2m′Z2(k′, iωm′) +

(�k′ − µ+ χ(k′, iωm′)

)2+ φ2(k′, iωm′)

(4)

χ(k, iωm) = −1

Nβ

∑k′,m′

λkk′(iωm − iωm′)g(�F )

�k′ − µ+ χ(k′, iωm′)ω2m′Z

2(k′, iωm′) +(�k′ − µ+ χ(k′, iωm′)

)2+ φ2(k′, iωm′)

(5)

along with the equation for the order parameter:

φ(k, iωm) =1

Nβ

∑k′,m′

[λkk′(iωm − iωm′)g(�F )

− Vk,k′] φ(k′, iωm′)ω2m′Z

2(k′, iωm′) +(�k′ − µ+ χ(k′, iωm′)

)2+ φ2(k′, iωm′)

. (6)

These are supplemented with the electron number equation, which determines the chemical potential, µ:

ne = 1−2

Nβ

∑k′,m′

�k′ − µ+ χ(k′, iωm′)ω2m′Z

2(k′, iωm′) +(�k′ − µ+ χ((k′, iωm′)

)2+ φ2(k′, iωm′)

. (7)

Written in this way both Z and χ are even functions ofiωm (and, as we’ve assumed from the beginning, they arealso even functions of k). With electron-phonon pairingthe anomalous self energy, which determines the anoma-lous pairing amplitude φ(k, iωm), is also an even func-tion of Matsubara frequency. A generalization of thislatter result, giving rise to so-called Berezinskii48 “odd-frequency” pairing, is beyond the scope of this review. Asurvey of Berezinskii pairing is given in Ref. [49].

Other symbols in Eqs. (4-7) are as follows.The numberof lattice sites is given by N , the parameter β ≡ 1/(kBT ),where kB is the Boltzmann constant and T is the tem-perature, µ is the chemical potential, and g(�F ) is theelectronic density of states at the Fermi level in the band.These equations are generally valid for multi-band sys-tems, and then the labels k and k′ are to be understood toinclude band indices. However, we shall proceed for sim-plicity with the assumption of a single band, with singleparticle energy �k. Because we are assuming finite tem-perature right from the start, the equations are writtenon the imaginary frequency axis, and are functions of theFermion Matsubara frequencies, iωm ≡ πkBT (2m − 1),with m an integer. Similarly the Boson Matsubara fre-quencies are given by iνn ≡ 2πkBTn, where n is an inte-ger. Finally, we have also included a direct Coulomb re-pulsion in the form of Vk,k′ , which in principle representsthe full (albeit screened) Coulomb interaction betweentwo electrons.

The key ingredient of Eliashberg theory (as opposedto BCS theory) is the presence of the electron-phononpropagator, contained in

λkk′(z) ≡∫ ∞

0

2να2kk′F (ν)

ν2 − z2dν (8)

with α2kk′F (ν) the spectral function of the phononGreen function. This function is sometimes written asα2kk′(ν)F (ν) to emphasize that the coupling part (α

2)can have significant frequency dependence. This spectralfunction is often called the Eliashberg function. Equa-tion (8) has been used as a “bosonic glue” to generalize

the application of the Eliashberg/BCS formalism to be-yond that of phonon exchange. Very often the boson isa collective mode of the very degrees of freedom that aresuperconducting, i.e. the conduction electrons. Exam-ples include spin fluctuations or plasmons, but this workis on more questionable footing.50

A significant anisotropy may exist, specifically throughthe nature of the coupling in the Eliashberg function.Since the important physical attribute of the Eliash-berg formalism beyond BCS is retardation, and there-fore in the frequency domain, we will nonetheless ne-glect anisotropy in what follows.51 More nuanced argu-ments for the wave vector dependence of the electron-phonon coupling are provided in Ref. [20], connected tothe energy scale hierarchy �F >> νphonon >> Tc, whereνphonon is a typical phonon energy scale (note that wehave adopted the standard practice of dropping h̄ andkB , and therefore we refer to temperatures and phononfrequencies as energies). Indeed very often the neglect ofanisotropy was justified by the study of so-called “dirty”superconductors, where the presence of impurities servedto self-average over anisotropies. We will also drop thewave vector dependence in the direct Coulomb repul-sion, although this step is less justified. It means thatthe direct Coulomb repulsion is represented by a singleparameter, which we will call U , since this is what wewould obtain by reducing the long-range Coulomb repul-sion with an on-site Hubbard interaction with strengthgiven by U . This is one of the weak points of the Eliash-berg description of superconductivity — an inadequatedescription of correlations due to Coulomb interactions.In what follows we will focus more attention on the re-tardation effects, since this is the part that Eliashbergtheory is best designed to handle properly.

Once we drop the wave vector dependence in the cou-pling function, all quantities (Z, χ and φ) become inde-pendent of wave vector. The integration over the FirstBrillouin Zone can then be performed, although here aseries of approximations are utilized. The result can leadto confusion, so we provide some detail here. First, once

4

it is determined that the unknown functions in Eqs. (4-7)do not depend on wave vector, we can replace the sumover wave vectors in the first Brillouin zone with an in-tegration over the electronic density of states,

1

N

∑k

→∫ �max�min

d� g(�), (9)

where g(�) is the single electron density of states and �minand �max are the minimum and maximum energies of theelectronic band. Since typically the energy scales are suchthat Tc > Tc. Now the equationsare

Z(iωm) = 1 +πTcωm

+∞∑m′=−∞

λ(iωm − iωm′)ωm′√

ω2m′ + ∆2(ωm′)

(15)

5

Z(iωm)∆(iωm) = πTc

+∞∑m′=−∞

[λ(iωm − iωm′)− u∗(ωc)θ(ωc − |ωm′ |)]∆(iωm′)√

ω2m′ + ∆2(ωm′)

, (16)

where u∗(ωc) ≡ g(�F )U∗(ωc). One should note thatU∗(ωc) < U , physically corresponding to the fact thatretardation effects allow two electrons to exchange aphonon with one another while not being at the sameplace at the same time. This means they do not feel thefull direct Coulomb interaction with one another.

Thus far we have written the Eliashberg equations asfunctions of imaginary frequency. As we will see in thenext subsection one can solve these equations as theyare, to determine many thermodynamic quantities of in-terest, in particular Tc. However, later we will extendthese equations to the upper half-plane, and in particu-lar just above the real axis. This is required for the eval-uation of dynamic quantities like the tunneling densityof states and the optical conductivity.17,34,36 In anticipa-tion of these results we note here that we use functionsZ(z) and φ(z) [and therefore ∆(z)] with the followingproperties53 as a function of complex frequency z

Z(z∗) = Z∗(z); Z(−z) = Z(z), (17)φ(z∗) = φ∗(z); φ(−z) = φ(z), (18)

∆(z∗) = ∆∗(z); ∆(−z) = ∆(z). (19)

A. Results on the imaginary axis: Tc

To compute actual results for Tc, along with the gapfunction ∆(ωm) and the renormalization function Z(ωm),we need to specify α2F (ν) (now assumed to be isotropic)and u∗(ωc). The latter quantity is very difficult to com-pute, and the former is more tractable through Den-sity Functional Theory. Historically it has been “mea-sured” through tunnelling measurements.18 We use quo-

tation marks around the word “measured” because infact the current is measured while the spectral func-tion is extracted through an inversion process that re-quires theoretical input through the Eliashberg equa-tions themselves.18 We will simply adopt a model spectralfunction given by

α2F (ν) =λ0ν02π

[�

(ν − ν0)2 + �2− �ν2c + �

2

]θ(νc−|ν−ν0|),

(20)that is, a Lorentzian line shape cut off in such a way thatthe function goes smoothly to zero in the positive fre-quency domain. This Lorentzian has a centroid given byν0 and a half-width given by �. The cutoff frequency pa-rameter νc makes the Lorentzian go to zero at frequencyν0 + νc and frequency ν0 − νc. For concreteness we willuse a variety of values of ν0 with � = 0, or � ≈ ν0/10. Thefirst choice results in a δ-function spectrum with weightsuch that the mass enhancement parameter, λ, definedby

λ ≡ 2∫ ∞

0

dνα2F (ν)

ν(21)

is simply given by λ0. As � increases λ decreases fromλ0; however in what follows we will adjust λ0 to keep λconstant.54 Since the main focus of Eliashberg theory isthe effect of retardation, we will often set u∗(ωc) = 0, butwe will nonetheless note how this quantity affects the gapfunction and Tc.

Superconducting Tc is determined by linearizing thegap equations, Eqs. (15,16) so that they become

Z(iωm) = 1 +πTcωm

{λ+ 2

m−1∑n=1

λ(iνn)

}. (22)

Z(iωm)∆(iωm) = πTc

+∞∑m′=−∞

[λ(iωm − iωm′)− u∗(ωc)θ(ωc − |ωm′ |)]∆(iωm′)

|ωm′ |. (23)

The latter of these two equations is an eigenvalue equa-tion and can be solved as such. We use a power methodthat iterates the eigenvalue and eigenvector simultane-ously by requiring that the gap function at the lowestMatsubara frequency,55 ∆(iω1), remain at unity. Thisprocedure tends to converge very quickly for stronger

coupling, but requires more care for weaker coupling.56

We begin with a standard plot of Tc vs λ in Fig. 1,for a simple δ-function spectral function with frequencyas shown. There are scaling relations for Tc with typi-cal phonon frequency, but we choose to show the resultsexplicitly in real units to make the result clear. The

6

0

20

40

60

80

100

120

140

0.0 0.2 0.4 0.6 0.8 1.0

Tc

(K)

λ

U = 0

ε = 0 (δ−function)

ν0 (meV)

100

100

1020

30

4050

Lorentzian, ε = 1 meV

u*(ωc = 1 eV) = 0.1

FIG. 1. The superconducting critical temperature, Tc (K)vs. the dimensionless mass enhancement parameter, λ, fora variety of characteristic phonon frequencies, as indicated.The solid curves are for the Einstein spectrum with U = 0.The square points indicate how Tc changes (for ν0 = 100 meVonly) when a Lorentzian is used instead with � = 10 meV andνc = 80 meV, according to Eq. (20). For the same valueof λ there is only a slight reduction in the value of Tc. Thepoints marked with asterisks are again for the same broadenedLorentzian centred at ν0 = 100 meV, but now with u

∗(ωc = 1eV. Note that νc is used as a practical cutoff for the phononspectrum whereas ωc is used for the Matsubara cutoff for thedirect Coulomb repulsion. See the discussion in the text forhow plausible these parameters might or might not be.

possibility of determining an expression for Tc analyt-ically has been discussed extensively in the literature20

and will not be done here. The trends are clear; higher Tccomes from higher values of λ and from higher values ofthe characteristic phonon frequency, which in the presentcase is provided by ν0. The width of the spectrum playsa minor role, and the Coulomb repulsion suppresses Tc,as indicated by the marked points. It is well known thatEliashberg theory predicts that Tc increases with bothfrequency and coupling strength as Tc ≈ ν0

√λ in the

asymptotic limit.24,57

B. Validity of the Theory

A perhaps more important question is the validity ofthe parameters used in the calculation. We have shownresults up to a value of λ = 1. Are higher values al-lowed? In particular, is it possible for a material to

have a sizeable value of electron-phonon coupling whilemaintaining a large phonon frequency? As discussed inthe introduction, this question has been the subject ofprevious investigation,40–42 although with only qualita-tive conclusions. The discovery of (very) high tempera-ture superconductivity in the hydrides58,59 under intensepressure has spurred a reassessment of this type of anal-ysis, since a part of the community believes that thesesuperconductors are electron-phonon driven. The mainevidence has been an observed isotope shift.58 Moreover,the prediction of superconductivity in some of these com-pounds through density functional theory calculations60

adds plausibility to this explanation. However, very highcharacteristic phonon frequencies (60 - 120 meV) andrather large electron-phonon coupling values (λ ≈ 2 ormore) are required. The latter is well outside the rangeconsidered reasonable, especially given that the charac-teristic phonon energy remains so high. Moreover, the su-perconductivity literature has unfortunately lapsed intosimply accepting as “standard” or “conventional” a valuefor the Coulomb pseudopotential u∗ = 0.1, and, espe-cially given the high value of phonon frequency, the an-ticipated reduction of the Coulomb interaction throughretardation will be much lower than previously thought,and the value of the direct Coulomb repulsion is undoubt-edly higher when the phonon frequency is so high.

An additional direction of addressing this questioncomes from microscopic calculations involving Quan-tum Monte Carlo (QMC) and Exact Diagonalization(ED) techniques, utilizing specific microscopic models.These methods provide controlled approximations andare therefore suitable for benchmarking more approxi-mate theories like Eliashberg theory. By far the mostwork in this direction has been done on the Holsteinmodel.61 The Holstein model retains only the short-range(on-site) interaction between local (Einstein) oscillatorsand the electron charge density. Because it is a very lo-cal model it is more amenable to the exact or controlledmethods developed over the past 40 years, and thereforeis a “favourite” for understanding the electron-phononinteraction, much like the Hubbard model62 is heavilyused for the study of electron-electron interactions. Ashort historical account of this activity is provided in theAppendix of Ref. [34].

Briefly, early Quantum Monte Carlo studies in onedimension63 and two dimensions64,65 established thatcharge-density-wave (CDW) correlations dominate athalf-filling and close to half-filling. The critical questionis whether, sufficiently away from half-filling, where thesusceptibility for superconductivity is stronger than thatfor CDW formation, do the “remnant” CDW correla-tions enhance or suppress suppress superconducting Tc?In Ref. [66] the present author, based on a comparisonof QMC and Migdal-Eliashberg calculations on (very!)small lattices, argued that CDW fluctuations actuallysuppress superconducting Tc. In the so-called renormal-ized Migdal-Eliashberg calculations a phonon self-energywas included; in this manner CDW fluctuations impacted

7

FIG. 2. The singlet pairing susceptibility vs. electron den-sity for the Holstein model on a 4 × 4 lattice. See Refs. [65and 66] for pertinent definitions. Here a bare dimensionlesscoupling strength λ0 = 2 and phonon (Einstein) frequencyωE = 1t are used, and the susceptibility is plotted for a tem-perature T = t/6, where t is the nearest neighbour hoppingparameter. In the topmost figure, QMC results are indicatedwith error bars. The solid curves are the result for Migdal-Eliashberg theory with a renormalized phonon propagatorand the dashed curves are the result for the unrenormalizedcalculations. The renormalized calculations agree very wellwith the QMC results (both done for a 4 × 4 lattice), in-dicating that this (combined Migdal-Eliashberg plus phononself-energy in the bubble RPA approximation) result accu-rately captures the impact of CDW fluctuations on the pairingsusceptibility. In the bottom figure the renormalized (solidcurve) and unrenormalized (dashed curves) are plotted fora larger system. The renormalized calculations stop at anelectron density close to n ≈ 0.9 because a CDW instabilityoccurs there. The unrenormalized calculations carry on tohalf-filling, because they are oblivious to the CDW instability(and fluctuations). This result indicates that CDW fluctu-ations at densities less than n = 0.9 suppress pairing, andpresumably Tc, even though λ

eff → ∞. Reproduced fromRef. [66].

the phonon propagator, resulting in softer phonons andan enhanced coupling constant. Comparisons with theQMC results served to benchmark the Eliashberg-likecalculations.

Figure 2, reproduced from Ref. [66], illustrates thatthe so-called renormalized calculations agree with theQMC results. These calculations (solid curves) includephonon self-energy effects which are essentially the CDWfluctuations.65 In contrast, the unrenormalized calcula-tions (dashed curves) are the standard Migdal-Eliashbergcalculations that omit phonon self-energy effects. Fig-ure 2(a) illustrates that the renormalized calculationsare more accurate, and Fig. 2(b) shows that includingCDW fluctuations suppresses the pairing susceptibility,χSP. We understand these results to indicate that in thevicinity of a CDW instability, while the effective couplingconstant (λeff in Ref. [65 and 66]) increases, supercon-ducting Tc actually decreases.

More recently similar calculations have beenperformed67 and other methodologies have beenemployed.68 In the latter reference the role of retarda-tion in reducing the direct Coulomb interaction was alsoaddressed; while they found qualitative agreement withthe standard arguments, quantitative agreement waslacking, especially for the expected large values of directCoulomb repulsion. An older calculation with just twoelectrons,69 based on Exact Diagonalization studies,also found qualitative support for a retardation-relatedreduction in the direct Coulomb repulsion. It is worthmentioning that finding this insensitivity to increasedCoulomb repulsion, known as the “pseudopotentialeffect,”70,71 has been looked for in QMC studies, butthese have mostly been unsuccessful. They may still bethere; part of the problem is that QMC results becomemore difficult as the electronic and phonon energyscales differ from one another by a significant amount.Moreover, it may be that if larger lattices and morerealistic phonon frequencies (i.e. significantly less thanthe electron hopping parameter, t) are used, the resultillustrated in Fig. 2 could change qualitatively.

An additional concern has been raised about theelectron-phonon coupling becoming too strong — thatof polaron collapse.43 Exact studies in the thermody-namic limit72 have established that a single electron, in-teracting with Einstein oscillators through the Holsteinmodel, acquires an additional mass which is modest forλ

8

Fermi sea.For example, in Migdal theory, the effective mass for

electrons near the Fermi energy is m∗/me ≈ 1 + λ, evenfor λ ≈ 2 or more, whereas a single electron with thiscoupling would have an effective mass many orders ofmagnitude higher. It is important to note that in thequantum treatment polarons never “self-localize,” essen-tially because of Bloch’s Theorem. However, with suchlarge effective mass ratios, any impurities (including sur-faces), would readily act as localization sites.

There are perhaps a few scenarios to work one’s wayout of this dilemma. First, as we have already mentioned,perhaps the Holstein model itself is pathological. Forthis reason it is important to study other models. Un-fortunately other models are more difficult to work with,but thus far the conclusions arrived at with the Holsteinmodel seem to hold for these other models as well. For ex-ample, in Ref. [76] a variational approach was used withthe Fröhlich Hamiltonian in the continuum with acous-tic phonons, and in Refs. [77 and 78] the Barisić-Labbé-Friedel/Su-Schrieffer-Heeger (BLF/SSH) model was ex-amined with perturbation theory and the adiabatic ap-proximation. In either case more definitive results asachieved with the Holstein model are still lacking, al-though all indications are that these models have strongpolaronic tendencies as well.

A second scenario is that as one assembles a Fermisea of polarons, they somehow become increasingly un-dressed, presumably due to some argument involvingPauli blocking. There are no calculations that we areaware of, however, that provide a demonstration of this.79

Part of the reason may be psychological; the Migdal ap-proximation is more often called the Migdal Theorem,and so one may be inclined to take it for granted thatthis is what will happen when we assemble a Fermi sea— the “theorem” will be fulfilled. However, in my opin-ion this is more a belief than an established argument,as the Migdal approximation does not foresee or accountfor polaron physics.

This discussion has been a long digression concern-ing the domain of applicability of Eliashberg theory, andclearly a lot more investigation is required on this ques-tion. For now we return to the properties of the solutionsto the Eliashberg equations.

C. Results on the imaginary axis: in thesuperconducting state

Returning to Eqs. (15,16), or their linearized counter-parts, Eqs. (22,23), once Tc is determined then the gapfunction can be determined both at Tc and below Tc.The gap function is a generalized (frequency-dependent)order parameter. It will grow continuously from zero atTc to its full value at zero temperature, but it dependson frequency. In Fig. 3(a) we show the gap functionas a function of Matsubara frequency for a variety oftemperatures, for λ = 1 and ν0 = 10 meV. Note that

0.0

0.5

1.0

1.5

2.0

2.5

0 20 40 60 80 100

∆(iω

m)

(meV

)

ωm (meV)

(a)

ν0 = 10 meVε = 0λ = 1U = 0

T/Tc = 0.1 0.5 0.7 0.8 0.9 0.95 0.98 0.99

0.0

0.2

0.4

0.6

0.8

1.0

0 20 40 60 80 100

∆(iω

m)/∆

(iω1)

ωm (meV)

(b)

ν0 = 10 meV

ε = 0λ = 1U = 0

T/Tc = 0.1 0.5 0.7 0.8 0.9 0.95 0.98 0.99 1.00

FIG. 3. (a) The gap function ∆(iωm) vs. Matsubara fre-quency, ωm, for various temperatures. We have used a δ-function phonon spectrum (� = 0) with ν0 = 10 meV,electron-phonon coupling strength, λ = 1 and U = 0. Forthese parameters, Tc = 13.3 K. Clearly the gap function in-creases in amplitude with decreasing temperature, and belowabout T/Tc = 0.5 there is very little change in the ampli-tude and in the frequency dependence. With a broadenedphonon spectrum there would be only minor changes. Witha nonzero U , the gap function would have a negative asymp-tote as ωm → ∞. In (b), to illustrate that there is very lit-tle change in the frequency dependence at all temperatureswe show the normalized gap function, ∆(iωm)/∆(iω1) vs.ωm. Now the results look very similar to one another, whichmakes convergence from one temperature to the next rela-tively easy. Also shown is the weak coupling expectation56 atTc, ∆(iωm)/∆(iω1) = ω

2E/(ω

2E + ω

2m), indicated with a solid

red curve. This result clearly does not resemble the data, sincewe the numerical results are for λ = 1. However, the slightlymodified result, ∆(iωm)/∆(iω1) = ω

2E/(ω

2E + ω

2m/(1 + λ)

2),shown as a dashed black curve, is a fairly good fit.

9

0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.2 0.4 0.6 0.8 1.0

∆(iω

1,T

)/∆

(iω1,

0)

T/Tc

ν0 = 10 meV

ε = 0

λ = 1.0

λ = 0.3

BCS

FIG. 4. The gap function at the first Matsubara frequency(serving as an order parameter), normalized to the zero tem-perature gap function at the first Matsubara frequency, vs. re-duced temperature T/Tc. The blue squares are the results ata few temperatures for calculations using a phonon δ-functionspectrum with ν0 = 10 meV, λ = 1, and U = 0. Shown forcomparison is the weak coupling BCS result (red curve). Thedeviations are very slight. Also shown for comparison are theEliashberg results for the same phonon spectrum but withλ = 0.3 (green triangles). These results fall right on the BCSweak coupling result.

these functions are defined on a discrete set of points(the Fermion Matsubara frequencies) that become moreclosely spaced as the temperature is lowered. For temper-atures close to Tc the gap function diminishes graduallyto zero at all frequencies, while at the lowest tempera-ture the gap function attains a maximum. In Fig. 3(b)we plot the normalized values, ∆(iωm)/∆(iω1) vs. Mat-subara frequency, and it is clear that they differ from oneanother by very little. Returning to Fig. 3(a), the lowestfrequency function value can be thought of as an orderparameter. In Fig. 4 we show the lowest frequency gapfunction value, ∆(iω1), now normalized to the value atT = 0 vs. reduced temperature, T/Tc. These are shownas blue squares, for about 9 temperatures. Also shown isthe BCS weak coupling result, given as a red curve. Onecan see that the differences are small. We have also aplotted many more points (green asterisks) for a weakercoupling, λ = 0.3 (same phonon frequency), which fallexactly on the BCS curve. The main point is that de-viations from the weak coupling BCS result are minor.For much stronger coupling than given here deviations

are similarly very small, and experiment confirms this tobe the case.80

Many measurable properties of the superconductingstate can be calculated from the imaginary axis solu-tions to the gap function. The renormalization function,Z(iωm), is also required but this does not change by verymuch in the superconducting state. Examples of measur-able properties include all thermodynamic quantities likethe specific heat, and various critical fields. Systematicchanges with coupling strength, as measured by λ, oralternatively the ratio of the critical temperature to aparticular phonon frequency moment, Tc/ωln, have beenreviewed elsewhere,33,34 and will not be repeated here.

D. Results on the real axis

For any dynamical property (tunneling current, op-tical response, dynamical penetration depth, etc.), therelevant Green function (and therefore self-energy) is re-quired as a function of real frequency. More precisely,for the retarded Green function it is needed at ω + iδ,i.e. infinitesimally above the real axis. In the originalliterature12,13,71,81,82 the spectral representation was in-troduced to replace Matsubara sums with real frequencyintegrals, and these equations were then solved, eitheranalytically (with approximations) or numerically. Thiswas a difficult task (especially with the computers avail-able at that time), and eventually the procedure wasadopted that first required a solution on the imaginaryaxis and then analytic continuation (iωm → ω + iδ)through some approximate process. For this type of an-alytic function, the method of Padé approximants wasused,83 although the degree of precision needed for thegap function on the imaginary axis was very stringent(10−12 for relative errors) in order to achieve accurateresults on the real axis.

An appreciation for the information imbedded in imag-inary axis solutions can be attained by considering thesimple example of g(iωm) = sech(ωm/ν0), a very smoothfunction without structure, and monotonically decreas-ing with frequency on the positive imaginary axis. Theanalytic continuation can be easily done analytically; itis g(ω + iδ) = sec[(ω + iδ)/ν0]. This function, in con-trast to its imaginary axis counterpart, is riddled withdivergences and discontinuities. And yet, in principle atleast, this information is embedded in the (smooth) re-sults on the imaginary axis. In practice, the informationis contained in the 10th significant digit and beyond.

An alternative, numerically exact procedure was devel-oped in the late 1980’s.84 Here we simply write down theresulting expressions, the derivation of which is availablein Ref. [84]. They are

10

φ(ω + iδ) = πT

∞∑m=−∞

[λ(ω − iωm)− u∗(ωc)θ(ωc − |ωm|)

] ∆(iωm)√ω2m + ∆

2(iωm)

+iπ

∫ ∞0

dν α2F (ν)

{[N(ν) + f(ν − ω)

] φ(ω − ν + iδ)√(ω − ν)2Z2(ω − ν + iδ)− φ2(ω − ν + iδ)

+[N(ν) + f(ν + ω)

] φ(ω + ν + iδ)√(ω + ν)2Z2(ω + ν + iδ)− φ2(ω + ν + iδ)

},

(24)

and

Z(ω + iδ) = 1 +iπT

ω

∞∑m=−∞

λ(ω − iωm)ωm√

ω2m + ∆2(iωm)

+iπ

ω

∫ ∞0

dν α2F (ν)

{[N(ν) + f(ν − ω)

] (ω − ν)Z(ω − ν + iδ)√(ω − ν)2Z2(ω − ν + iδ)− φ2(ω − ν + iδ)

+[N(ν) + f(ν + ω)

] (ω + ν)Z(ω + ν + iδ)√(ω + ν)2Z2(ω + ν + iδ)− φ2(ω + ν + iδ)

}, (25)

and of course ∆(ω + iδ) ≡ φ(ω + iδ)/Z(ω + iδ). Heref(ω) ≡ 1/(exp(βω) + 1) and N(ν) ≡ 1/(exp(βν)− 1) arethe Fermi and Bose functions respectively. Note that incases where the square–root is complex, the branch withpositive imaginary part is to be chosen. The reason forthis can be traced back to Eq. (6) [or Eq. (4)] where theintegration over �k′ (with the assumptions made there)requires that the pole (given the same square–root thatappears here) be above the real axis.

It can easily be verified that substituting ω+ iδ → iωninstantly recovers the imaginary axis equations (all theFermi and Bose factors cancel to give zero contribu-tions beyond the initial terms that require Matsubarasummations). Clearly the inverse is not true — replac-ing the Matsubara frequency iωm where it appears inEqs. (15,16) produces the first lines in Eqs. (24,25) involv-ing Matsubara sums, but leaves out the remaining twolines in each case. The strategy for solving these equa-tions is straightforward; the imaginary axis equations[Eqs. (15,16) ] are first solved self-consistently. Theseare then used in Eqs. (24,25) and these equations areiterated to convergence. The presence of the first linesin these equations provides a “driving term” that makesthe iteration process quite rapid. For example, perform-ing the entire operation (solution of imaginary axis andreal axis equations) for a given temperature takes abouta tenth of a second on a laptop.

Moreover, T = 0 is a special case, as is clear fromthese equations. In fact, this was recognized a long timeago,85 where they established the following low frequency

behaviour at T = 0,

Re∆(ω + iδ) = c,

Im∆(ω + iδ) = 0,T = 0

ReZ(ω + iδ) = d,

ImZ(ω + iδ) = 0.(26)

where c and d are constants. In contrast, the behaviourat any non-zero temperature is

Re∆(ω + iδ) ∝ ω2,Im∆(ω + iδ) ∝ ω,

T > 0

ReZ(ω + iδ) = d(T ),

ImZ(ω + iδ) ∝ 1/ω.(27)

For conventional parameter choices this distinction hasvery little consequence, as the differences are barely dis-cernible. For example, the expression of the imaginarypart of Z(ω + iδ) in the normal state is given by

ImZ(ω + iδ) = 2πλν0ω

[N(ν0) + f(ν0)] (28)

for a δ-function spectrum [� → 0 in Eq. (20)] withstrength λ and central frequency ν0. For ν0 = 10 meVand λ = 1 then Fig. 1 indicates Tc ≈ 10 K, and atT/Tc = 0.1, the exponent in the Bose and Fermi func-tions makes this part ≈ 10−50. Thus, true to Eq. (27)the limiting behaviour is ∝ ν0/ω. However, to see thisrequires ω/ν0 < 10

−45, making it unobservable, and in-distinguishable from the T = 0 case.

While this expression is for the normal state, the corre- sponding one of the superconducting state is even more

11

−3

−2

−1

0

1

2

3

4

5

6

0 10 20 30 40 50 60

Re ∆

(ω +

iδ)

(meV

)

ω (meV)

(a)T/Tc = 0.10 0.50 0.60 0.70 0.80 0.90 0.95 0.99

ν0 = 10 meVνc = 8 meVε = 1 meVλ = 1.0 −1

0

1

2

3

4

5

0 10 20 30 40 50 60

Im ∆

(ω +

iδ)

(meV

)

ω (meV)

(b)T/Tc = 0.10

0.50

0.60

0.70

0.80

0.90

0.95

0.99

0.0

0.5

1.0

1.5

2.0

2.5

3.0

0 10 20 30 40 50 60

Re Z

(ω +

iδ)

ω (meV)

(c)T/Tc = 0.10 0.50 0.60 0.70 0.80 0.90 0.95 0.99 1.00

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

0 10 20 30 40 50 60

Im Z

(ω +

iδ)

ω (meV)

(d)T/Tc = 0.10 0.50 0.60 0.70 0.80 0.90 0.95 0.99 1.00

0

2

4

6

8

10

0 2 4 6 8 10 12 14

g(ω

)/g(ε

F)

ω (meV)

α2F(ν)

ν (meV)

(e)

0

1

2

0 5 10 15 20

FIG. 5. Frequency dependence of (a) Re ∆(ω+ iδ), (b) Im ∆(ω+ iδ), (c) Re Z(ω+ iδ), (d) Im Z(ω+ iδ), and (e) g(ω)/g(�F ),for various temperatures in the superconducting state. Features are discussed in the text. For the spectral function we haveused Eq. (20) with λ = 1, ν0 = 10 meV, and � = 1 meV. It is displayed in the inset of (e) and was used for all these figures.The colour coding in (e) is the same as the others. We used U = 0.

severe, due to the development of the superconductingorder parameter, which gives rise to a gap in the spec-trum. Again, given the first two lines in Eq. (27) thereis technically no gap, but in practice for reasons like wehave just indicated, the practical results more closely fol-low the behaviour indicated in Eq. (26). Where the finite

temperature result becomes pronounced and noticeablydifferent than the zero temperature behaviour is in thestrong coupling limit,86 but in this case the parametersare not realistic and undoubtedly beyond the domain ofvalidity of the theory.

Returning to T = 0, for cases like the present where

12

the phonon spectrum has a gap there is a special sim-plification. Basically, no iteration is required — thelow frequency gap and renormalization functions comeentirely from the first lines of Eqs. (24,25), and thesecan be constructed explicitly from the imaginary axis re-sults. For a δ-function phonon spectrum, however, onehas to be careful to convert the Matsubara summationto an actual integral, as a discontinuity will occur at thephonon frequency (at non-zero temperature this discon-tinuity is broadened into a gradual drop). Once the lowfrequency gap and renormalization functions are so con-structed, higher frequency values require the Matsubarasum and real axis values of these functions at lower fre-quencies only. Eventually the entire functional forms areso constructed, without the need for iteration. As we willsee, low temperature results converge quite rapidly to thezero temperature result, so this non-iterative method canbe used as an alternative, for the lowest temperatures.Nonetheless, we will proceed with fully converged (itera-tive) finite temperature results, since these require suchfew iterations anyways.

To show real axis results, we utilize a phonon spectrumas in Eq. (20) with ν0 = 10 meV, νc = 8 meV, and� = 1 eV, and λ = 1. Results with a δ-function spectrumtend to have a series of singularities, that are anywaysartifacts of the singular spectrum, so we prefer to showresults corresponding to this model spectrum. A series ofsuch plots was also shown in Ref. [34] for the tunneling-derived Pb spectrum, with a much larger value of λ, andmany more such results have been shown in the literature,often using this same method.87 In Fig. 5 we show (a) thereal part of the gap function, (b) the imaginary part ofthe gap function, (c) the real part of the renormalizationfunction, (d) the imaginary part of the renormalizationfunction, and (e) the tunneling density of states,

g(ω)

g(�F )= Re

ω√ω2 −∆2(ω + iδ)

, (29)

which is measurable in single-particle tunneling experi-ments. The first observation, difficult to make with justthese results, is that an image of the α2F (ν) spectrumis contained in both the real and imaginary parts of thegap function. Here it is the peak structure clearly evi-dent in (a) centred around 10 meV for the highest tem-perature shown. As the temperature decreases this peakshifts to higher frequency, roughly by an amount equalto the value of the gap function at low frequency (about2 meV in the present case). Experimentation with differ-ent spectral functions makes this observation more self-evident. See, for example, the distinctive spectrum forPb in Fig. 4.35(a) of Ref. [34].

Both functions in (a) and (b) go to zero as the criti-cal temperature is approached from below. As discussedearlier, they both go to zero at zero frequency at all tem-peratures shown, according to Eqs. (27), although onecannot see this on the scale shown. Even for the highesttemperatures shown this behaviour can barely be seen,but becomes evident when one expands the low frequency

scale. For the lowest temperatures shown even expandingthe frequency scale by a few orders of magnitude is notenough to reveal the low-frequency behaviour indicatedby Eqs. (27). For this reason one cannot use ∆(ω + iδ)as an order parameter at any frequency; either one hasto revert to φ(ω + iδ) at zero frequency, or one can usethe imaginary axis result for ∆(iωm), as we did in Fig. 4.Also note that these functions approach zero at high fre-quency. If a Coulomb repulsion is included then the realpart of ∆(ω+ iδ) approaches a negative constant at highfrequency.

In contrast the real and imaginary parts of Z(ω + iδ)plotted in (c) and (d) have changed very little in thesuperconducting state, and remain non-zero at the su-perconducting critical temperature, as indicated by theblack curve. An image of α2F (ν) is present in this func-tion as well, particularly in the imaginary part (see alsoFig. 4.35 (c) and (d) in Ref. [34]). Finally, the tunnel-ing density of states is shown in Fig. 5(e), and revealsa “gap” that opens from zero at T = Tc rather quicklyand then saturates to a low temperature value as indi-cated. In fact a plot of this “gap” vs. temperature wouldfollow the result displayed in Fig. 4 very closely. How-ever, as first pointed out by Karakozov et al.85 there isno “gap” (hence the parentheses) and in fact this is evi-dent in Fig. 5(e), where there is a noticeable rounding ofthe curves at frequencies below the sharp peak at almostall temperatures shown. The peak is a remnant of thesquare-root singularity known from BCS theory, whichis evident from Eq. (29) if a constant gap function isused, ∆(ω+ iδ) = ∆0. In fact Eliashberg theory predictssmearing of this singularity simply due to the presenceof imaginary components of all the functions involved inEqs. (27) for all finite temperatures. It is also worthpointing out that the BCS limit of Eliashberg theory isnot achieved by setting the gap function to a constant,∆(ω+ iδ)→ ∆0, but in fact the gap function is a decay-ing function of frequency in this limit.85 This frequencydependence and its implications for the weak couplinglimit has been further explored in Refs. [56, 88, and 89].

Finally, though not so evident in Fig. 5(e), there are“ripples” in the density of states beyond the “gap” region,caused by the coupling of electrons to phonons. The pres-ence of these ripples in experiments (see, in particular,Refs. [18, 90, and 91]) is perhaps the strongest evidenceof the validity of Eliashberg theory. In fact the most in-tense scrutiny has been superconducting Pb, where theelectron-phonon coupling is particularly strong, λ ≈ 1.5,with a value well beyond the expected domain of validity.These experiments, coupled with an inversion techniquethat use the Eliashberg equations themselves, result in aconsistent description of the superconducting state for Pband other so-called “strong coupling” superconductors.

Many systematic deviations from BCS theory havebeen characterized, for example the gap ratio,2∆0/(kBTc),

92 the specific heat jump, and many otherdimensionless ratios93,94 These have all been reviewed inRef. [33], and show very systematic behaviour as a func-

13

tion of the strong coupling index, Tc/ωln. On the otherhand, when systematics are examined with purely exper-imental parameters, the picture is not so clear.95

III. SUMMARY AND OUTLOOK

I have provided just a sketch of what we consider theessence of Eliashberg theory — retardation effects, in thecontext of a single featureless band. The generalizationof these types of calculations to more complicated scenar-ios is well documented in a number of places, and havenot been reviewed here. These include order parame-ter anisotropy, multi-band superconductivity, Berezinskii“odd-frequency” pairing, sharply varying electronic den-sity of state, impurity effects, and so on. These addi-tional complications are increasingly taken into accountto understand new classes of compounds that exhibit su-perconductivity, such as the hydrides, MgB2, and thepnictides. In some cases, these additional effects havebeen invoked to explain higher critical temperatures aswell, but for the most part they are motivated by match-ing theory to experiment.

In its bare form, Fig. 1 presents the possibilities for Tcprovided by Eliashberg theory. The conscious decisionwas made to extend the domain of coupling strength tounity only and not beyond, because there are reasons tobelieve that going beyond this regime is not viable. Atthe same time, large values of the characteristic phononfrequency have been used, and this is why the plot ex-tends to beyond ≈ 50 K for the vertical axis, Tc. Arethese values of frequency, together with large values ofλ ≈ 1 viable? Probably not, but given these sorts ofparameter values, this is what Eliashberg in its standardform predicts.

I have also tried to touch on aspects of the theory wheremore critical scrutiny is possible, by comparing results tothose obtained with microscopic models, such as the Hol-stein model. We believe there are significant difficultiesthat arise when these comparisons are made. One re-action is to dismiss such comparisons, as the Holsteinmodel (or the Hubbard model, for that matter) may beregarded as “toy models,” possibly fraught with patholo-gies. However, if the Holstein model is lacking in someway, it is important to know why, and what other aspectof the electron-phonon interaction (wave-vector depen-dence?) is essential to the success of Eliashberg theory.For example, if the super-high-Tc of the hydrides is con-firmed to originate in the electron-phonon interaction,then clearly one or more missing gaps in our understand-ing of how this happens needs to be filled.

Moreover, as presented here, Eliashberg theory fo-cusses on the superconducting instability, and does notconsider other, possibly competing, or potentially en-hancing, instabilities. This possibility has come up asmore and more phase diagrams of families of materials ex-hibit a nearby antiferromagnetic or charge-density-waveinstability, as a function of some tuning parameter (dop-

ing, pressure, etc.). It would be desirable to generalizeEliashberg theory to be more “self-regulating,” and havethe theory itself indicate when a competing instability islimiting superconducting Tc, for example.

The other aspect that goes hand in hand with theelectron-phonon coupling is the direct Coulomb interac-tion. We cannot claim to understand superconductivityto the point of having predictive power until we under-stand the role of Coulomb correlations, and their detri-mental (or perhaps favourable?) effects on pairing. Akey advancement has to come in further understandingthe role that competitive tendencies or instabilities playin superconductivity. Many of the modern-day meth-ods (Dynamical Mean Field Theory, for example) seek toaddress the question of competing interactions. Studieswith Quantum Monte Carlo methods, like the ones men-tioned here, will also aid in furthering our understandingof interacting electrons, and similar studies with the now-accessible much larger lattice sizes would be welcome.

ACKNOWLEDGMENTS

This work was supported in part by the NaturalSciences and Engineering Research Council of Canada(NSERC). I would like to thank the many students andpostdoctoral fellows in my own group who have con-tributed to the work described here. I also want tothank in particular Jules Carbotte, who first taught meabout Eliashberg theory, and Ewald Schachinger, whowas instrumental in teaching me about programming theEliashberg imaginary axis equations. I also want to thankSasha Alexandrov, who over the years continued to ques-tion the validity of MIgdal-Eliashberg theory. In thesame way, Jorge Hirsch, with whom I have worked agreat deal on other matters, has been instrumental indiscussions concerning the validity of the work reviewedhere. I also want to thank him for critical comments con-cerning parts of this review. Finally, Jules sadly passedaway earlier this year (April 5, 2019), and this review isdedicated to his memory. He was a wonderful man, and Ifeel extremely fortunate to have first entered the physicsworld under his guidance.

APPENDIX: Derivation of Eliashberg Theory

In this Appendix, we will first outline a derivation ofEliashberg theory, based on a weak coupling approach.Our primary source for this derivation is Ref. [47]. Migdaltheory of the normal state follows by simply dropping theanomalous amplitudes in what follows.

If we know the many-body wave function of system,we can calculate the expectation value for any observ-able. However, usually this is something we do not know,and instead we calculate multi-electron Green functions,which are themselves related to observables. The Greenfunctions are necessarily almost always approximate, andthose computed in Eliashberg theory are no exception. In

14

fact, Eliashberg theory is essentially a mean-field theory,though because of the inherent frequency dependence inthe self energy, it is in many ways a precursor to Dynam-ical Mean Field Theory.96

We begin with the definition of the one-electron Greenfunction, defined in momentum space,97 as a function ofimaginary time, τ ,

G(k, τ − τ ′) ≡ − < Tτ ckσ(τ)c†kσ(τ′) >, (30)

where k is the momentum and σ is the spin. The an-gular brackets denote a thermodynamic average. Wecan Fourier-expand this Green function in imaginary fre-quency:

G(k, τ) =1

β

∞∑m=−∞

e−iωmτG(k, iωm)

G(k, iωm) =

∫ β0

dτG(k, τ)eiωmτ . (31)

The frequencies iωm are the Fermion Matsubara frequen-cies, given by iωm = iπT (2m − 1), m = 0,±1,±2, ...,where T is the temperature. Because the c’s are Fermionoperators, the Matsubara frequencies are odd multiplesof iπT. The imaginary time τ takes on values from 0 toβ ≡ 1/(kBT ).

A similar definition holds for the phonon Green func-tion,

D(q, τ − τ ′) ≡ − < TτAq(τ)A−q(τ ′) >, (32)

where

Aq(τ) ≡ aq(τ) + a†−q(τ). (33)

The Fourier transform is similar to that given in (31)except that the Matsubara frequencies are iνn ≡ iπT2n,n = 0,±1,±2, ... and occur at even multiples of iπT .These are the Boson Matsubara frequencies.

To derive the Eliashberg equations, we follow Ref. [47],and use the equation-of-motion method. The startingpoint is the time derivative of Eq. (30),

∂

∂τG(k, τ) = −δ(τ) − < Tτ

[H−µN, ckσ(τ)

]c†kσ(0) >,

(34)where we have put τ ′ = 0, without loss of gener-ality. We use the Hamiltonian (2), and assume, forthe Coulomb interaction, the simple Hubbard model,HCoul = U

∑i ni↑ni↓. Including the Coulomb repulsion,

the result is repeated here,

H =∑kσ

�kc†kσckσ

+∑q

h̄ωqa†qaq

+1√N

∑kk′σ

g(k,k′)(ak−k′ + a

†−(k−k′)

)c†k′σckσ

+U

N

∑k,k′,q

c†k↑c†−k+q↓c−k′+q↓ck′↑, (35)

where the various symbols have already been defined inthe text. We consider only the Green function with σ =↑;the commutator in Eq. (34) is straightforward and weobtain(

∂

∂τ+ �k

)G↑(k, τ) =

−δ(τ)− 1√N

∑k′

gkk′ < TτAk−k′(τ)ck′↑(τ)c†k↑(0) >

+U

N

∑pp′

< Tτ c†p′−k+p↓(τ)cp′↓(τ)cp↑(τ)c

†k↑(0) > . (36)

Various higher order propagators now appear; to deter-mine them another equation of motion can be written,which would, in turn, generate even higher order prop-agators, and this eventually leads to an infinite set ofequations with hierarchical structure. This infinite se-ries is normally truncated at some point by the processof decoupling, which is simply an approximation proce-dure. For example, in (36) the Coulomb term is normallydecoupled at this point and becomes

< Tτ c†p′−k+p↓(τ)cp′↓(τ)cp↑(τ)c

†k↑(0) >→

< Tτ c†p′−k+p↓(τ)cp′↓(τ) >< Tτ cp↑(τ)c

†k↑(0) >→

−δkpG↓(p′, 0)G↑(k, τ). (37)

The case of the electron–phonon term is more difficult;in this case we define a ‘hybrid’ electron/phonon Greenfunction,

G2(k,k′, τ, τ1) ≡< TτAk−k′(τ)ck′↑(τ1)c†k↑(0) >, (38)

and write out an equation of motion for it. We simplyget

∂

∂τG2(k,k

′, τ, τ1) = −ωk−k′ < TτPk−k′(τ)ck′↑(τ1)c†k↑(0) >,(39)

where Pq(τ) = aq(τ)− a−q(τ). Taking a second deriva-tive yields[∂2

∂τ2− ωk−k′

]G2(k,k

′, τ, τ1) =∑k′′σ

2ωk−k′gk−k′ < Tτ c†k′′−k+k′σ(τ)ck′′σ(τ)ck′↑(τ1)c

†k↑(0) > .

(40)

At this point we do not simply decouple the last line ofEq. (40). We first need to take the phonon propaga-tor into account, and the standard procedure is to usethe “non-interacting” phonon propagator. The adjective“non-interacting” is in quotes because part of the phi-losophy of proceeding in this way was a desire to notcompute corrections to the phonon propagator, becausethe information going into this part of the calculations(e.g. the phonon spectral function) was going to comefrom experiment. Coming from experiment means that

15

nature “had already done the calculation,” and we didnot want to double count. Clearly, if the purpose of thiscalculation is to compare to Quantum Monte Carlo cal-culations where this is not the case, then something dif-ferent should be done, and this is what motivated therenormalized Migdal-Eliashberg calculations of Refs. [65and 66].

For now, we proceed with the standard Eliashberg cal-culations. The equation of motion for the non-interactingphonon propagator is(

∂2

∂τ2− ω2q

)D(q, τ − τ ′) = 2ωqδ(τ − τ ′). (41)

Inserting this expression into Eq. (40) then yields

G2(k,k′, τ, τ) =

1

N

∑k′′σ

∫ β0

dτ ′gk′′,k′′+k−k′D(k− k′, τ − τ ′)

× < Tτ c†k′′σ(τ′)ck′′+k−k′σ(τ

′)ck′↑(τ)c†k↑(0) >, (42)

where now τ1 has been set equal to τ as is requiredin (36). It is important that this be done only afterapplying Eq. (41). The result can be substituted intoEq. (36), and then Fourier transformed (from imaginarytime to imaginary frequency). Before doing this however,we recall that Gor’kov11 realized the important role ofthe so-called Gor’kov anomalous amplitude, in the Wickdecomposition97 of the various two–particle Green func-tions encountered above. We therefore have to accountfor these in addition to the pairing of fermion operatorsused in Eq. (37).

The anomalous amplitudes are defined to be

F (k, τ) ≡ − < Tτ ck↑(τ)c−k↓(0) > (43)

and

F̄ (k, τ) ≡ − < Tτ c†−k↓(τ)c†k↑(0) > . (44)

Now we need to repeat the same steps as above with Fand F̄ as we did with G. Skipping the intermediate steps,

the result is an equation analogous to Eq. (36)

(∂

∂τ− �k

)F̄ (k, τ) =

− 1√N

∑k′

g−k′,−k < TτAk−k′(τ)c†−k′↓(τ)c

†k↑(0) >

+U

N

∑k′′,q

< Tτ c†k′′↑(τ)c

†−k′′+q↓(τ)ck+q↑(τ)c

†k↑(0) >,

(45)

and similarly for the function F . This leads to the needfor another ‘hybrid’ electron/phonon anomalous Greenfunction,

F̄2(k,k′, τ, τ1) ≡< TτAk−k′(τ)c†−k′↓(τ1)c

†k↑(0) >, (46)

and, following the same process as for the regular Greenfunction, we find

F̄2(k,k′, τ, τ) =

1

N

∑k′′σ

∫ β0

dτ ′gk′′,k′′+k−k′D(k− k′, τ − τ ′)

× < Tτ c†k′′σ(τ′)ck′′+k−k′σ(τ

′)c†−k′↓(τ)c†k↑(0) >, (47)

where again τ1 has been set equal to τ after applyingEq. (41).

The Fourier definitions of the anomalous Green func-tion are the same as Eq. (31):

F̄ (k, τ) =1

β

∞∑m=−∞

e−iωmτ F̄ (k, iωm)

F̄ (k, iωm) =

∫ β0

dτF̄ (k, τ)eiωmτ , (48)

and similarly for F . In frequency space one finds thattwo self energies naturally arise,

Σ(k, iωm) = −1

Nβ

∑k′,m′

gkk′gk′kD(k− k′, iωm − iωm′)G(k′, iωm′), (49)

φ(k, iωm) = −1

Nβ

∑k′,m′

gk′kg−k′−kD(k− k′, iωm − iωm′)F (k′, iωm′). (50)

One can show that gk′k = g∗kk′ ; normally one expects a similar relation with negative wave vectors, and we assume it

in what follows. These equations are then written self-consistently and lead to Eqs. (4-7) once Eq. (3) is used.

1 See, for example, Superconducting Quantum Levitation ona 3π Möbius Strip.

2 See, for example, Applications with Superconductivity.

https://www.youtube.com/watch?v=Vxror-fnOL4https://www.youtube.com/watch?v=Vxror-fnOL4http://www.supraconductivite.fr/en

16

3 R.D. Parks, in the Preface of Superconductivity, edited byR.D. Parks (Marcel Dekker, Inc., New York, 1969) page v.

4 This phrase was first used by an anonymous referee in ap-praising a colleague’s understanding of the subject matterof his grant proposal (in 1990). I have plagiarized it eversince.

5 See the various articles in the Special Issue of PhysicaC, ‘Superconducting Materials: Conventional, Unconven-tional and Undetermined’, edited by J.E. Hirsch, M.B.Maple, and F. Marsiglio, Physica C 514, 1-444 (2015).

6 See, for example, P. P. Kong, V. S. Minkov, M. A. Ku-zovnikov, S. P. Besedin, A. P. Drozdov, S. Mozaffari,L. Balicas, F.F. Balakirev, V. B. Prakapenka, E. Green-berg, D. A. Knyazev, M. I. Eremets, ‘Superconductivityup to 243 K in yttrium hydrides under high pressure,’ inhttps://arxiv.org/abs/1909..

7 Danfeng Li, Kyuho Lee, Bai Yang Wang, Motoki Osada,Samuel Crossley, Hye Ryoung Lee, Yi Cui, Yasuyuki Hikitaand Harold Y. Hwang, ‘Superconductivity in an infinite-layer nickelate’, Nature 572, 624 (2019).

8 As summarized in F. London, In: Superfluids Volume I:Macroscopic Theory of Superconductivity (John Wiley andSons, New York, 1950).

9 V. L. Ginzburg and L. D. Landau, Zh. Eksperim. i. Teor.Fiz. 20, 1064 (1950).

10 J. Bardeen, L.N. Cooper and J.R. Schrieffer, MicroscopicTheory of Superconductivity, Phys. Rev. 106, 162 (1957);Theory of Superconductivity Phys. Rev. 108, 1175 (1957).

11 L.P. Gor’kov, On the energy spectrum of superconductors,Zh. Eksperim. i Teor. Fiz. 34 735 (1958); [Sov. Phys. JETP7 505 (1958)]. Zh. Eksperim. i Teor. Fiz. 34 735 (1958);[Sov. Phys. JETP 7 505 (1958)].

12 G.M. Eliashberg, Interactions between Electrons and Lat-tice Vibrations in a Superconductor, Zh. Eksperim. i Teor.Fiz. 38 966 (1960); Soviet Phys. JETP 11 696-702 (1960).

13 G.M. Eliashberg, Temperature Green’s Function for Elec-trons in a Superconductor, Zh. Eksperim. i Teor. Fiz. 381437-1441 (1960); Soviet Phys. JETP 12 1000-1002 (1961).Nambu independently formulated the same theory.15

14 Based on Web of Science, Oct. 1, 2019. These numbers area little high, as a (very) few of these pertain to a mathe-matician with the same name.

15 Yoichiro Nambu, Quasi-Particles and Gauge Invariance inthe Theory of Superconductivity Phys. Rev. 117 648-663(1960).

16 Based on Web of Science, Oct. 1, 2019. These numbers area little high, as a (very) few of these pertain to a mathe-matician with the same name.

17 D.J. Scalapino, The Electron-Phonon Interaction andStrong-coupling Superconductivity, In: Superconductivity,edited by R.D. Parks (Marcel Dekker, Inc., New York,1969)p. 449.

18 W.L. McMillan and J.M. Rowell, Tunneling and Strong-coupling Superconductivity, In: Superconductivity, editedby R.D. Parks (Marcel Dekker, Inc., New York, 1969)p.561.

19 Superconductivity in Two Volumes, edited by R.D. Parks(Marcel Dekker, Inc., New York, 1969) p. 1-1412.

20 P.B. Allen and B. Mitrović, Theory of Superconducting Tc,in Solid State Physics, edited by H. Ehrenreich, F. Seitz,and D. Turnbull (Academic, New York, 1982) Vol. 37, p.1.

21 C.S. Owen and D.J. Scalapino, S-state instabilities for re-tarded interactions, Physica 55 691 (1971).

22 G. Bergmann and D. Rainer, The sensitivity of the transi-

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23 D. Rainer and G. Bergmann, Temperature dependence ofHc2 and κ1 in strong coupling superconductors, J. LowTemp. Phys. 14 501 (1974).

24 P.B. Allen and R.C. Dynes, Transition temperature ofstrong-coupled superconductors reanalyzed, Phys. Rev. B12905 (1975).See also R.C. Dynes, McMillan’s equation and the Tc ofsuperconductors, Solid State Commun. 10, 615 (1972).

25 J.M. Daams and J.P. Carbotte, Thermodynamics of strongcoupling superconductors including the effect of anisotropy,J. Low Temp. Phys. 43 263 (1981).

26 D. Rainer, Principles of Ab Initio Calculations of Su-perconducting Transition Temperatures, Progress in LowTemperature Physics Volume 10, edited by D.F. Brewer(North-Holland, 1986), Pages 371-424.

27 J.P. Carbotte and R.C. Dynes, Superconductivity in SimpleMetals, Phys. Rev. 172, 476-484 (1968).

28 Lilia Boeri, Understanding Novel Superconductors with AbInitio Calculations, In: Andreoni W., Yip S. (eds) Hand-book of Materials Modeling. Springer, Cham. pp. 1-41.

29 Antonio Sanna, Introduction to Superconducting DensityFunctional Theory E. Pavarini, E. Koch, R. Scalettar,and R. Martin (eds.) The Physics of Correlated Insula-tors, Metals, and Superconductors Modeling and Simula-tion Vol. 7 Forschungszentrum Julich, 2017, ISBN 978-3-95806-224-5

30 F. Giustino, Electron-phonon interactions from first prin-ciples, Rev. Mod. Phys. 89, 015003 (2017).

31 M. Lüders, M. A. L. Marques, N. N. Lathiotakis, A. Floris,G. Profeta, L. Fast, A. Continenza, S. Massidda, and E. K.U. Gross, Ab initio theory of superconductivity. I. Densityfunctional formalism and approximate functionals, Phys.Rev. B 72, 024545 (2005).

32 M. A. L. Marques, M. Lüders, N. N. Lathiotakis, G. Pro-feta, A. Floris, L. Fast, A. Continenza, E. K. U. Gross,and S. Massidda, Ab initio theory of superconductivity. II.Application to elemental metals, Phys. Rev. B 72, 024546(2005).

33 J.P. Carbotte, Properties of boson-exchange superconduc-tors, Rev. Mod. Phys. 62, 1027-1157 (1990).

34 F. Marsiglio and J.P. Carbotte, ‘Electron-Phonon Super-conductivity’, Review Chapter in Superconductivity, Con-ventional and Unconventional Superconductors, edited byK.H. Bennemann and J.B. Ketterson (Springer-Verlag,Berlin, 2008), pp. 73-162. Note that an earlier version ofthis review was published by the same editors in 2003, butthe author order was erroneously inverted, and one of theauthor affiliations was incorrect in that version.

35 W. Lee, D. Rainer and W. Zimmermann, Holstein effectin the far-infrared conductivity of high Tc superconductors,Physica C 159, 535-544 (1989).

36 F. Marsiglio and J.P. Carbotte, Aspects of Optical Proper-ties in Conventional and Oxide Superconductors, Aust. J.Phys. 50, 975 (1997);Quasiparticle Lifetimes and the Conductivity ScatteringRate Aust, J. Phys. 50, 1011 (1997).

37 Giovanni A.C. Ummarino, Eliashberg Theory, In EmergentPhenomena in Correlated Matter Modeling and SimulationVol. 3 Forschungszentrum Julich, 2013, ISBN 978-3-89336-884-6, edited by E. Pavarini, E. Koch, and U. Schollwock.

38 A.B. Migdal, Interaction between electrons and lattice vi-brations in a normal metal, Zh. Eksp. Teor. Fiz. 34, 1438

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(1958) [Sov. Phys. JETP 7, 996 (1958)].39 W.L. McMillan, Transition Temperature of Strong-Coupled

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41 Jonathan E. Moussa and Marvin L. Cohen, Two bounds onthe maximum phonon-mediated superconducting transitiontemperature, Phys. Rev. B 74, 094520 (2006).

42 I. Esterlis, S. A. Kivelson and D. J. Scalapino, A bound onthe superconducting transition temperature, npj QuantumMaterials (2018) 3:59, p1-4.

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47 G. Rickayzen, In: Theory of Superconductivity (John Wileyand Sons, New York, 1965).

48 V. L. Berezinskii, New model in the anistotropic phase ofsuperfluid He-3, Pisma Zh. Eksp. Teor. Fiz. 20, 628 (1974).

49 J. Linder and A.V. Balatsky, Reviews of Modern Physics,accepted, Sept. 2019.

50 Having said this, in the spirit of phenomenological explo-ration even the present author has used this formalism toexplore other types of “bosonic glue.” See, for example, S.Verga, A. Knigavko and F. Marsiglio, Inversion of angle-resolved photoemission measurements in high-Tc cupratesPhys. Rev. B67, 054503 (2003).

51 This is not without irony, as one of the most celebratedelectron-phonon-driven superconductors is MgB2, with aTc ≈ 39 K. Ab initio calculations have explained this byfinding considerable anisotropy in the coupling function.See Hyoung Joon Choi, David Roundy, Hong Sun, MarvinL. Cohen and Steven G. Louie, The origin of the anomaloussuperconducting properties of MgB2, Nature 418, 758-760(2002).Not everyone agrees with this interpretation. See, for ex-ample, J.E. Hirsch and F. Marsiglio, Electron-phonon orhole superconductivity in MgB2, Phys. Rev. B 64, 144523(2001).

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53 Vinay Ambegaokar and Ludwig Tewordt, Theory of theElectronic Thermal Conductivity of Superconductors withStrong Electron-Phonon Coupling, Phys. Rev. 134, A805(1964).

54 We could have elected to keep the area,∫∞

0dνα2F (ν) con-

stant as well, for example.55 There are undoubtedly references that describe this pro-

cedure in more detail, but I learned it primarily from amentor at the time, Ewald Schachinger, who was a fre-quent visitor at McMaster University during the course ofmy PhD.

56 F. Marsiglio, Eliashberg theory in the weak-coupling limit,

Phys. Rev. B98, 024523 (2018).57 D. Rainer first obtained the asymptotic result noted in the

text in 1973 (private communication).58 A. P. Drozdov, M. I. Eremets, I. A. Troyan, V. Kseno-

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59 Maddury Somayazulu, Muhtar Ahart, Ajay K. Mishra,Zachary M. Geballe, Maria Baldini, Yue Meng, ViktorV. Struzhkin, and Russell J. Hemley, Evidence for Super-conductivity above 260 K in Lanthanum Superhydride atMegabar Pressures, Phys. Rev. Lett. 122, 027001 (2019).

60 D. Duan, Y. Liu, F. Tian, D. Li, X. Huang, Z. Zhao, H.Yu, B. Liu, W. Tian, and T. Cui, Pressure-induced metal-lization of dense (H2S)2H2 with high-Tc superconductivity,Sci. Rep. 4, 6968 (2014).

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